Moessner's theorem

In number theory, Moessner's theorem or Moessner's magic is related to an arithmetical algorithm to produce an infinite sequence of the exponents of positive integers $1^{n},2^{n},3^{n},4^{n},\cdots ~,$ with $n\geq 1~,$ by recursively manipulating the sequence of integers algebraically. The algorithm was first published by Alfred Moessner in 1951; the first proof of its validity was given by Oskar Perron that same year.

For example, for $n=2$, one can remove every even number, resulting in $(1,3,5,7\cdots )$, and then add each odd number to the sum of all previous elements, providing $(1,4,9,16,\cdots )=(1^{2},2^{2},3^{2},4^{2}\cdots )$.

Construction

Write down every positive integer and remove every $n$-th element, with $n$ a positive integer. Build a new sequence of partial sums with the remaining numbers. Continue by removing every $(n-1)$-st element in the new sequence and producing a new sequence of partial sums. For the sequence $k$, remove the $(n-k+1)$-st elements and produce a new sequence of partial sums.

The procedure stops at the $n$-th sequence. The remaining sequence will correspond to $1^{n},2^{n},3^{n},4^{n}\cdots ~.$

Example

The initial sequence is the sequence of positive integers,

$1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16\cdots ~.$

For $n=4$, we remove every fourth number from the sequence of integers and add up each element to the sum of the previous elements

$1,2,3,5,6,7,9,10,11,13,14,15\cdots \to 1,3,6,11,17,24,33,43,54,67,81,96\cdots$

Now we remove every third element and continue to add up the partial sums

$1,3,11,17,33,43,67,81\cdots \to 1,4,15,32,65,108,175,256\cdots$

Remove every second element and continue to add up the partial sums

$1,15,65,175\cdots \to 1,16,81,256\cdots$,

which recovers $1^{4},2^{4},3^{4},4^{4},\cdots$.

Variants

If the triangular numbers are removed instead, a similar procedure leads to the sequence of factorials $1!,2!,3!,4!,\cdots ~.$

External links

• Polster, Burkard (17 July 2021). The Moessner Miracle. Why wasn't this discovered for over 2000 years?. Mathologer (short video documentary). Retrieved July 20, 2021 – via YouTube.